Local Geometric Geoid Models Parameters and Accuracy Determination Using Least Squares Technique

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Local Geometric Geoid Models Parameters and Accuracy Determination Using Least Squares Technique Eteje Sylvester Okiemute Ph. D. Candidate, Department of Surveying and Geoinformatics, Nnamdi Azikiwe University, Awka, Nigeria Oduyebo Fatai Olujimi Ph. D. Candidate, Department of Surveying and Geoinformatics, Nnamdi Azikiwe University, Awka, Nigeria Abstract: The absence of national local geoid model in some countries has led to the determination of local geoid model in various parts of those countries. Local geoid models are determined using the geometric and gravimetric methods amongst others. Using the geometric method requires fitting an interpolation surface to points of known geoidal undulations which requires the determination of the geometric geoid model parameters and its accuracy using least squares technique. Because of the rigorous as well as the matrix nature of the technique, researcher have been experiencing difficulty in its application for the determination of geometric geoid models’ parameters and their accuracy. This paper presents a detailed procedure for the determination of geometric geoid models’ parameters as well as their accuracy using least squares technique. The steps to be considered when applying the technique are enumerated in sequential order. The enumerated steps were also demonstrated with a numerical example.

Keywords: Geometric geoid, model parameters, accuracy, least squares

1. Introduction Least squares are a statistical method used to determine a line of best fit by minimizing the sum of squares created by a mathematical function. It is a popular method for determining regression equations. Instead of trying to solve an equation exactly, least squares method is used to determine a close approximation which is known as the estimate. Modelling methods that are often used when fitting a function to a curve include the straight-line method, polynomial method, logarithmic method and Gaussian method. The Least-Squares criterion is an imposed condition for obtaining a unique solution for an incompatible system of linear equations. The term adjustment, in a statistical sense, is a method of deriving estimates for random variables from their observed values. The application of the least-squares criterion in the adjustment problem is called the LeastSquares Adjustment method (Mohammad-Karim, 1981). The method of least squares is a rigorous technique that can be applied to the adjustment of horizontal geodetic network to yield the most likely values of the survey measurements. In geodesy, it is desirable or necessary to fit a plane or curve surface to a set of points with known coordinates or heights. In solving this type of problem, it is first necessary to decide on the appropriate functional model for the data as stated by Ghilani (2010). The decision as to whether to use a plane or curve surface depends on the size of the application area. To determine the best fit surface, two or more surfaces have to be applied and the one with smaller residuals after least squares solution with the surfaces selected.

Geometric geoid models are surfaces that are fitted to the geoidal undulations of an area to enable geoid heights of new points within the area to be interpolated. These surfaces are plane as well as curve surfaces depends on the degree. The curve surfaces are ether quadratic or polynomial in nature. The plane surfaces are usually applied in small areas while the curve surfaces are applied in relatively large areas. The larger the area the higher the order as well as the degree of the polynomial model/surface. To apply any of these models in a particular area, the model parameters as well as its accuracy have to be determined using least squares technique. Obtaining the accuracy of the model enables the reliability of the model to be determined.

Various researchers have been experiencing difficulty in the application of least squares adjustment technique for determination of geometric geoid models’ parameters and their accuracy. The difficulty in its application resulted from its matrix nature. The computation of these parameters cannot be handled by Least squares adjustment software as the model terms are not obtained directly from measurement, that is, they are not bearings, azimuth, angles, distances, change in INTERNATIONAL JOURNAL OF INNOVATIVE RESEARCH & DEVELOPMENT

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northing and easting. They are normally reduced from the variables. Also, the dimension of the coefficient matrix depends on the number of points heights and the order as well as the degree of the model. The general matrix notation of least squares adjustment is simply the sum of the estimate and the matrix of observations equals to the residual matrix. In geoid model parameters determination, the residual is not truly needed. Making the matrix notation of least squares adjustment for geometric geoid parameters determination to be simply observation equal the estimate. Previous studies in which least squares adjustment technique was applied for the determination of local geometric geoid models never presented the breakdown of how the technique was applied in the studies. To determine the reliability of the model, the geoid heights of points from the model are compared with their known geoid heights.

This paper presents step by step application of observation equation method of least squares adjustment technique for determination of local geometric geoid models’ parameters and their accuracy.

1.1. Observation Equation Method of Least Squares Adjustment Equations that relate observed quantities to both observational residuals and independent unknown parameters are called observation equations. One equation is written for each observation and for a unique set of unknowns. For a unique solution of unknowns, the number of equations must equal the number of unknowns. Usually, there are more observations (and hence equations) than unknowns, and this permits determination of the most probable values for the unknowns based on the principle of least squares (Ghilani and Wolf, 2006).

Ayeni (2001) and Okwuashi and Asuquo (2014) explained that, in the observations equation method, the adjusted observations are expressed as a function of the adjusted parameter. The functional relationship between adjusted observations and the adjusted parameters as given in Ono et al (2014) is:

La = F ( X a ) Where,

(1)

La = adjusted observations and X a = adjusted parameters. Equation (1) is linear function and the general

observation equation model was obtained.

To make the matrix expression for performing least squares adjustment, analogy will be made with the systematic procedures. The system of observation equations is presented by matrix notation as (Mishima and Endo, 2002 and Ono et al, 2018):

V = AX − L

(2)

But the residual matrix is not necessary when applying least squares adjustment technique for the determination of local geometric geoid model parameters. Thus, the general matrix notation becomes

L = AX

(3)

where, A = Design Matrix, X = Vector of Unknowns, L = Observation Matrix. That is,

 a11  a A =  21 ...  a  m1

a12 a 22 ... am 2

... a1n   ... a 2 n  , ... ...   ... a mn 

 x1    x  X = 2  ...   x   m

and

 l1    l  L= 2  ...   l   m

The determination of the unknown parameters, X, requires the normal matrix, N and the matrix of numeric terms, t to be deduced. It is to be noted here that the observations are not weighted.

According to Ghilani (2010), a system of unweighted linear observation equations can be expressed in matrix notation as:

A T AX = A T L

(4)

To make X the subject of the formula, both sides of equation (4) will be divided by

T

A A . Thus,

X = ( AT A) −1 AT L (5) T T If A A = N , normal matrix and A L = t , matrix of numeric terms, then equation (5) becomes (6) X = N − 1t 1.2. Accuracy/Reliability of Geometric Geoid Model The accuracy of determined local geometric geoid model is obtained using the Root Mean Squares Error, RMSE index.

To evaluate the determined local geometric geoid model accuracy, the local geoid model is used to determine the geoidal heights of points whose geoid heights are known. The geometric geoid model geoidal undulations are compared with the INTERNATIONAL JOURNAL OF INNOVATIVE RESEARCH & DEVELOPMENT

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known geoidal undulations of the points to obtain the residuals. The Root Mean Squares Error, RMSE index for the computation of geometric geoid model accuracy as given by Kao et al (2017) is

RMSE = ±

V TV n

(7)

Where,

V = N KNOWN − N MODEL (Residual) N KNOWN = Known geoid height of point N MODEL = Model geoid height of point n = Number of points 1.3. Bicubic Geoid Model The bicubic surface is one of the polynomial geometric geoid models and it is given as (8) N = a o + a1 x + a 2 y + a 3 x 2 + a 4 y 2 + a 5 xy + a 6 x 2 y + a 7 xy 2 + a8 x 3 + a 9 y 3 , Where, y = ABS (Y − Yo )

(9)

x = ABS ( X − X o )

Y = Northing coordinates of observed station X = Easting coordinates of observed station Yo = Northing coordinates of the origin (average of the northing coordinates)

X o = Easting coordinates of the origin (average of the easting coordinates) 2. Steps to Be Considered When Computing Geometric Geoid Models Parameters and Accuracy Using Observation Equation Method of Least Squares Adjustment Technique The steps to be considered when computing geometric geoid models’ parameters and their accuracy using the least squares method are as follow: • Deduce the coefficient matrix, A, observation matrix, L and matrix of unknown parameters, X from the given or model of interest. It is to be noted here that the geometric geoid model is already established. The coefficients of the model terms on one side of the equation form the matrix of unknown parameters while the terms form the coefficient matrix.

The model terms which form the coefficient matrix are deduced from the given variables. The observation matrix is deduced from the term on the other side of the equation as well as the model. The number of points heights deduced with respect to the centroid of the study area determines the number of observation equations. In this, the weights of the observed heights are not truly necessary.

• Having deduced the above stated matrices, the model parameters are computed using equations (6).

• Since the parameters are computed, the next step is to substitute the computed parameters accordingly in the model.

• Having substituted the computed parameters in the model, it (the model) can now be used to develop a program such that the geoid heights of new points in the study area can be determined if the variables of the new points are given.

• The model accuracy has to be computed using equation (7) to determine its reliability. To determine the model accuracy, the geoidal heights of points obtained from the model are compared with their corresponding known geoid heights to obtained the residuals.

2.1. Numerical Application in Bicubic Geoid Model Parameters and Accuracy Determination The ellipsoidal heights of eight benchmarks whose rectangular coordinates and orthometric heights are known were determined using DGPS. The rectangular coordinates, orthometric heights, ellipsoidal heights and the geoid heights as computed from the orthometric and ellipsoidal heights are given in table 1. To interpolate geoid heights of new points, the bicubic geoid model, equation (8) is to be fitted to the computed geoid heights of the benchmarks. Compute the model parameters and accuracy using least squares technique.

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Station

Coordinates Orthometric Ellipsoidal Height, H Height, h Northing Easting (m) (m) 249308.287 354033.425 175.189 209.237 244533.051 278026.486 291.686 326.581 249241.822 362785.077 222.300 256.677 259174.974 355889.303 425.449 460.033 247210.935 359597.719 325.386 359.665 260606.174 332700.238 120.829 155.115 252751.094 344865.087 143.546 177.828 276864.558 374129.027 315.314 349.689 243587.154 340245.247 257.359 291.966 269356.441 361478.369 199.075 233.531 256457.248 357864.254 351.273 385.448 Table 1: Coordinates and Heights of Benchmarks

ESO 01 ESO 02 ESO 03 ESO 04 ESO 05 ESO 06 ESO 07 ESO 08 ESO 09 ESO 10 ESO 11

Solution Using equation (9),

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Geoidal Height, N (m) 34.048 34.895 34.377 34.584 34.279 34.286 34.282 34.375 34.607 34.456 34.175

Yo and X o are computed as the mean as well as the centroid of the given positions as:

Yo = 255371.976

X o = 347419.476 Deduction of normal equations using equation (8)

N 1 = a o + a1 x1 + a 2 y1 + a 3 x12 + a 4 y12 + a5 xy1 + a 6 x 2 y1 + a 7 xy12 + a8 x13 + a9 y13 N 2 = a o + a1 x 2 + a 2 y 2 + a 3 x 22 + a 4 y 22 + a5 xy 2 + a 6 x 2 y 2 + a 7 xy 22 + a8 x 23 + a 9 y 23 N 3 = a o + a1 x3 + a 2 y 3 + a 3 x 32 + a 4 y 32 + a5 xy 3 + a 6 x 2 y 3 + a 7 xy 32 + a8 x 33 + a9 y 33 N 4 = a o + a1 x 4 + a 2 y 4 + a 3 x 42 + a 4 y 42 + a5 xy 4 + a 6 x 2 y 4 + a 7 xy 42 + a8 x 43 + a 9 y 43 N 5 = a o + a1 x5 + a 2 y 5 + a3 x 52 + a 4 y 52 + a 5 xy 5 + a 6 x 2 y 5 + a 7 xy 52 + a8 x 53 + a9 y 53 N 6 = a o + a1 x 6 + a 2 y 6 + a 3 x 62 + a 4 y 62 + a 5 xy 6 + a 6 x 2 y 6 + a 7 xy 62 + a8 x 63 + a 9 y 63 N 7 = a o + a1 x 7 + a 2 y 7 + a 3 x 72 + a 4 y 72 + a5 xy 7 + a 6 x 2 y 7 + a 7 xy 72 + a8 x 73 + a 9 y 73 N 8 = a o + a1 x8 + a 2 y 8 + a 3 x82 + a 4 y 82 + a5 xy8 + a 6 x 2 y 8 + a 7 xy 82 + a8 x83 + a9 y 83 N 9 = a o + a1 x9 + a 2 y 9 + a3 x92 + a 4 y 92 + a5 xy 9 + a 6 x 2 y 9 + a 7 xy 92 + a8 x 93 + a9 y 93 N 10 = a o + a1 x10 + a 2 y10 + a 3 x102 + a 4 y102 + a5 xy10 + a 6 x 2 y10 + a 7 xy102 + a8 x103 + a9 y103 N 11 = a o + a1 x11 + a 2 y11 + a 3 x112 + a 4 y112 + a 5 xy11 + a 6 x 2 y11 + a 7 xy112 + a8 x113 + a 9 y113 Also using equation (3), the coefficient matrix, A, matrix of unknown parameters, X and observation matrix, L are:

1  1 1  1  1 A = 1  1 1  1 1  1 

x1 x2

y1 y2

x 12 x 22

y 12 y 22

xy 1 xy 2

x 2 y1 x2 y2

xy 12 xy 22

x 13 x 23

x3

y3

x 32

y 32

xy 3

x 2 y3

xy 32

x 33

x4

y4

x 42

y 42

xy 4

x2 y4

xy 42

x 43

x5 x6

y5 y6

x 52 x 62

y 52 y 62

xy 5 xy 6

x 2 y5 x 2 y6

xy 52 xy 62

x 53 x 63

x7

y7

x 72

y 72

xy 7

x 2 y7

xy 72

x 73

x8 x9

y8 y9

x x

y y

x y8 x 2 y9

xy xy

x

y

xy 10

x 2 y 10

xy

x 11

y 11

x

xy 11

x 2 y 11

xy

2 8 2 9 2 10 2 11

x 83 x 93

y 10

2 8 2 9 2 10 2 11

xy 8 xy 9

x 10

2 8 2 9 2 10 2 11

y

2

x 103 x 113

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y 13   y 23  y 33  y 43   y 53  , y 63   y 73  y 83   y 93  y 103  y 113 

 N 1   34.048   ao         N 2   34.895   a1  N     a2  34.377  3      N 4   34.584   a3  N     a4   5   34.279  L =  N 6  =  34.286  , X =  a   5     N a  34.282  7    6 34.375 N  8     a7   N   34.607  9  a8         N 10   34.456   a9     34.175  N    11  DOI No. : 10.24940/ijird/2018/v7/i7/JUL18098

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Considering equation (3), the above matrices are rearranged as

L

× X.

A

=

 N 1  1 x1    N 2  1 x 2  N  1 x 3  3   N 4  1 x 4  N  1 x 5  5    N 6  = 1 x6    N 7   1 x 7  N 8  1 x 8 N    9  1 x 9  N 10  1 x 10    N 1 x  11   11

2 1 2 2 2 3 2 4 2 5 2 6 2 7 2 8 2 9 2 10 2 11

2 1 2 2 2 3 2 4 2 5 2 6 2 7 2 8 2 9 2 10 2 11

2

2 1 2 2 2 3 2 4 2 5 2 6 2 7 2 8 2 9 2 10 2 11

3 1 3 2 3 3 3 4 3 5 3 6 3 7 3 8 3 9 3 10 3 11

3 1 3 2 3 3 3 4 3 5 3 6 3 7 3 8 3 9 3 10 3 11

y1 y2 y3

x x x

y y y

xy1 xy 2 xy 3

x y1 x 2 y2 x 2 y3

xy xy xy

x x x

y y y

y4 y5 y6

x x x

y y y

xy 4 xy5 xy 6

x 2 y4 x 2 y5 x 2 y6

xy xy xy

x x x

y y y

y7 y8 y9

x x x

y y y

xy 7 xy8 xy 9

x 2 y7 x 2 y8 x 2 y9

xy xy xy

x x x

y y y

y10 y11

x x

y y

xy10 xy11

x 2 y10 x 2 y11

xy xy

x x

y y

  a    o   a1    a2       a3   a  × 4    a5   a    6   a7   a    8   a9   

Deduction of values of x and y for each equation using equation (9) and the computation of coefficient matrix, A elements as presented in table 2.

S/N

ao

1

1

2

1

3

1

4

1

5

1

6

1

7

1

8

1

9

1

10

1

11

1

x 6613.

949 69392 .990 15365 .601 8469.

827 12178 .243 14719 .238 2554.

389 26709 .551 7174.

229 14058 .893 10444 .778

y 6063.

689 10838 .925 6130.

154 3802.

998 8161.

041 5234.

198 2620.

882 21492 .582 11784 .822 13984 .465 1085.

272

x2 y2 xy x 2y xy2 43744326 367683 401049 265251997 243183849 .185 26.494 33.205 452.369 614.972 48153870 117482 752145 521936195 815244796 10.672 299.097 422.811 30177.900 3697.900 23610170 375787 941935 144733985 577420711 5.266 90.293 05.455 5871.900 367.821 71737975 144627 322107 272819364 122497354 .570 92.405 34.984 572.745 867.272 14830961 666025 993871 121036084 811102588 1.424 93.171 45.613 6490.330 289.749 21665595 273968 770434 113402013 403260404 6.596 26.800 01.522 5309.530 149.425 6524901.

686902 669475 171009975 175461553 306 3.411 1.662 69.926 43.902 71340013 461931 574057 153328107 123379717 4.047 073.211 218.010 50101.30 26395.30 51469556 138882 845470 606559571 996371464 .527 033.857 08.771 445.613 373.840 19765248 195565 196606 276406418 274943108 2.610 256.251 099.626 9287.230 3264.540 10909339 117781 113354 118395987 123020157 5.065 4.919 23.605 214.349 85.887 Table 2: Computation of Coefficient Matrix, Elements

x3 28932275 8332.3 33415410 0926675.0 36278446 84395.0 60760826 8493.0 18061505 41087.3 31890105 10466.4 16667133 748.4 19054597 523146.2 36925436 6335.649 27787751 76073.390 11394563 32395.640

y3 222951703 593.398 127338185 0100.930 230363778 461.463 550019679 61.395 543546505 687.124 143400411 060.929 180029010 64.308 992809138 5353.1400 163670007 3254.860 273487544 5700.330 127824933 8.993

Computation of the model parameters, X using equation (6) Using coefficient matrix, A as well as the computed coefficient matrix, A elements given in table 2 and the observation matrix, L, the model parameters are computed as:

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 a o   34.859077901082436       a1   0.000225556443422   a   − 0.000501234804656    2   a3   − 0.000000030132587     0.000000003398167  a  X =  4  =  a 5   0.000000051102750     0.000000000004046    a6    a 7   − 0.000000000008676   a    8   − 0.000000000000149  a     9   0.000000000004302  Substitution of the computed parameters into the interpolation model Having computed the model parameters, they are substituted into the geometric geoid model. Thus,

N = 34.859077901082436 + 0.000225556443422 x − 0.000501234804656 y − − 0.000000030132587 x 2 + 0.000000003398167 y 2 + 0.000000051102750 xy + + 0.000000000004046 x 2 y − 0.00000000000867 xy 2 − 0.000000000000149 x 3 + + 0.000000000004302 y 3 Computation of the model accuracy using equation (7) The local geometric geoid model whose parameters have been determined was used to obtain the geoid heights of the points and compared with the known geoidal heights of the points to obtain the residual as shown in table 3.

Known Geoidal Model Geoidal Difference in Geoid Height, ΔN Height, N (m) Height, N (m) (Residual) ESO 01 34.048 34.047 0.001 (m) ESO 02 34.895 34.948 -0.053 ESO 03 34.377 34.376 0.001 ESO 04 34.584 34.584 0.000 ESO 05 34.279 34.281 -0.002 ESO 06 34.286 34.289 -0.003 ESO 07 34.282 34.282 0.000 ESO 08 34.375 34.384 -0.009 ESO 09 34.607 34.608 -0.001 ESO 10 34.456 34.457 -0.001 ESO 11 34.175 34.175 0.000 Table 3: Known and Model Geoidal Heights and Residuals

Station

 0.001     - 0.053   0.001     0.000   - 0.002  T   , V = (0.001 - 0.053 0.001 0.000 - 0.002 - 0.003 0.000 - 0.009 - 0.001 - 0.001 0.000) V =  - 0.003     0.000   - 0.009     - 0.001   - 0.001     0.000 

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 0.001     - 0.053  0.001     0.000   - 0.002   V TV = (0.001 - 0.053 0.001 0.000 - 0.002 - 0.003 0.000 - 0.009 - 0.001 - 0.001 0.000) ×  - 0.003 = 0.003    0.000   - 0.009    - 0.001  - 0.001    0.000 

RMSE = Accuracy = ± The computed local geometric geoid model accuracy is the application area with accuracy of ± 17mm.

0.003 = ±0.0165m 11

± 0.017m.This implies that geoid heights can be interpolated within

3. Conclusion Considering that fitting a geometric geoid surface to a set of points whose geoidal heights are known to enable the interpolation of the geoid heights of new points if the variables are known requires the determination of the model parameters using the least squares technique which is rigorous and difficult to apply. This paper has presented the step by step procedures to be followed when determining geometric geoid model parameters and its accuracy using the least squares technique. The procedures were demonstrated using a numerical example. Considering the detailed procedures and the numerical example, it is certain that the difficulty in the application of least squares adjustment technique in local geometric geoid model parameters and accuracy determination has been simplified.

4. References i. Ayeni, O. O. (2001): Statistical Adjustment and Analysis of Data. A Manual, in the Department of Surveying & Geoinformatics, University of Lagos, Nigeria. ii. Ghilani, C. D. (2010): Adjustment Computations: Spatial Data Analysis. Fifth Edition. John Wiley & Sons, Inc., Hoboken, New Jersey. iii. Ghilani, C. D. and Wolf, P. R. (2006): Adjustment Computations: Spatial Data Analysis.

Fourth Edition. John Wiley & Sons, Inc., Hoboken, New Jersey. iv. Kao, S., Ning, F., Chen, C. and Chen, C. (2017): Using Particle Swarm Optimization to Establish a Local Geometric Geoid Model. Boletim de Ciências Geodésicas, Vol. 23, No. 2, pp. 327-337. v. Mishima, K. and Endo, K. (2002): The Method of the Design for Survey Network by Q Matrices. Proceedings of the 7th International Workshop on Accelerator Alignment, Spring. vi. Mohammad-Karim, M. (1981): Diagrammatic Approach to Solve Least-Squares Adjustment and Collocation Problems.

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DOI No. : 10.24940/ijird/2018/v7/i7/JUL18098

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